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Image restoration and enhancement methods that respect important features such as edges play a fundamental role in digital image processing. In the last decades a large
variety of methods have been proposed. Nevertheless, the correct restoration and
preservation of, e.g., sharp corners, crossings or texture in images is still a challenge, in particular in the presence of severe distortions. Moreover, in the context of image denoising many methods are designed for the removal of additive Gaussian noise and their adaptation for other types of noise occurring in practice requires usually additional efforts.
The aim of this thesis is to contribute to these topics and to develop and analyze new
methods for restoring images corrupted by different types of noise:
First, we present variational models and diffusion methods which are particularly well
suited for the restoration of sharp corners and X junctions in images corrupted by
strong additive Gaussian noise. For their deduction we present and analyze different
tensor based methods for locally estimating orientations in images and show how to
successfully incorporate the obtained information in the denoising process. The advantageous
properties of the obtained methods are shown theoretically as well as by
numerical experiments. Moreover, the potential of the proposed methods is demonstrated
for applications beyond image denoising.
Afterwards, we focus on variational methods for the restoration of images corrupted
by Poisson and multiplicative Gamma noise. Here, different methods from the literature
are compared and the surprising equivalence between a standard model for
the removal of Poisson noise and a recently introduced approach for multiplicative
Gamma noise is proven. Since this Poisson model has not been considered for multiplicative
Gamma noise before, we investigate its properties further for more general
regularizers including also nonlocal ones. Moreover, an efficient algorithm for solving
the involved minimization problems is proposed, which can also handle an additional
linear transformation of the data. The good performance of this algorithm is demonstrated
experimentally and different examples with images corrupted by Poisson and
multiplicative Gamma noise are presented.
In the final part of this thesis new nonlocal filters for images corrupted by multiplicative
noise are presented. These filters are deduced in a weighted maximum likelihood
estimation framework and for the definition of the involved weights a new similarity measure for the comparison of data corrupted by multiplicative noise is applied. The
advantageous properties of the new measure are demonstrated theoretically and by
numerical examples. Besides, denoising results for images corrupted by multiplicative
Gamma and Rayleigh noise show the very good performance of the new filters.

This paper presents a new similarity measure and nonlocal filters for images corrupted by multiplicative noise. The considered filters are generalizations of the nonlocal means filter of Buades et al., which is known to be well suited for removing additive Gaussian noise. To adapt to different noise models, the patch comparison involved in this filter has first of all to be performed by a suitable noise dependent similarity measure. To this purpose, we start by studying a probabilistic measure recently proposed for general noise models by Deledalle et al. We analyze this measure in the context of conditional density functions and examine its properties for images corrupted by additive and multiplicative noise. Since it turns out to have unfavorable properties for multiplicative noise we deduce a new similarity measure consisting of a probability density function specially chosen for this type of noise. The properties of our new measure are studied theoretically as well as by numerical experiments. To obtain the final nonlocal filters we apply a weighted maximum likelihood estimation framework, which also incorporates the noise statistics. Moreover, we define the weights occurring in these filters using our new similarity measure and propose different adaptations to further improve the results. Finally, restoration results for images corrupted by multiplicative Gamma and Rayleigh noise are presented to demonstrate the very good performance of our nonlocal filters.